Vibrations that can be described by the Wave Equation which is often applicable in slender continua. Examples of this include the vibrations of a string, axial vibrations of a rod or torsional vibrations. Long slender continua are especially susceptible to torsional vibrations because of the small ratio of diameter to length, in particular when torques are transferred via the continuum. This occurs in many types of technical equipment, for example, with long drive shafts. A particularly extreme case occurs with deep-hole drill strings used for drilling for gas or oil but also for geothermal projects. The total string reaches lengths of several kilometers so the ratio of diameter to length is often smaller than that of a human hair due to the fact that the outside diameter is only a few centimeters. FIG. 1 shows schematically the structure of a deep-hole drill string. The drill string is driven by a top drive actuator placed on the upper end of the string, for example. The so-called drill bit is located at the lower end of the string, i.e., an industrial diamond-tipped drill bit, which crushes the rock. Strong torsional vibrations, so-called stick-slip vibrations may occur in the string due to torques acting externally along the string, but in particular because of the nonlinear friction characteristic occurring between the rock and the drill bit. These effects are manifested in the drill bit coming to a standstill while the drive continues to rotate at a constant speed. This causes severe twisting of the string until the force on the bit becomes so big that the bit breaks loose again. The speed of the bit after breaking loose often reaches twice the amount of the drive speed and the string is being rotated in the other direction beyond its equilibrium position. As a result the drill bit again comes to a standstill. These vibrations are undesirable because they slow down the drilling operation and result in additional heavy loads on the drill rods.
Controlling these torsional vibrations has long been a topic of research in the field of mechanics. All the approaches so far in an attempt to control torsional vibrations have always been characterized by at least one of the following disadvantages.
On the one hand, measurements along the entire drill string must be available. On the basis of these measurements, the active modes of the drill string dynamics may be determined. Using the resulting modal representation, there are then various approaches for damping the torsional vibrations. Examples from the literature include E. Kreuzer and O. Kust, Analysis of long torsional strings by proper orthogonal decomposition, Archive of Applied Mechanics 67 (1996), no. 1, 68-80, and E. Kreuzer and M. Steidl, A Wave-Based Approach to Adaptively Control Self-Excited Vibrations in Drill-Strings, published in Proceedings of Applied Mathematics and Mechanics, 2010. In Kreuzer, Steidl, which constitutes the state of the art so far at the Institute of Mechanics and Ocean Engineering, the momentary active modes are converted into traveling waves to compensate the traveling waves at the top drive. To do so, first of all measurements along the entire drill string are necessary, secondly, continuous control is impossible and instead only a feedforward control to stabilize the string is possible. This method is not suitable if the drill string is unstable in the range around the desired target speed.
On the other hand, the dynamics of the drill string is not completely known. Therefore, the control cannot be tailored for the momentary system performance, and accordingly, the methods function better or worse, depending on the actual dynamics. The literature in this regard includes J. D. Jansen and L. Van den Steen, Active damping of self-excited torsional vibrations in oil well drillstrings, Journal of Sound and Vibration 179 (1995), 647-668, and R. W. Tucker and C. Wang, On the effective control of torsional vibrations in drilling systems, Journal of Sound and Vibration 224 (1999), 101-122. Various sources mention that the so-called “impedance control system” or “soft torque system” presented by Jansen and Van den Steen, which uses measurements of the motor current and motor voltage to implement the characteristic of a passively attenuated vibration absorber with the help of the actuator, is currently in use. The approach presented by Tucker and Wang uses measurements of the “contact torque” between the drill string and the top drive. Some frequencies are absorbed better with this method than others.
Singular disturbances, for example, a wave front caused by breaking loose, could not be controlled with such systems known from the state of the art.